calculate the ratio of wavelength o mth line of lyman series and balmer series of H-atom

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To calculate the ratio of the wavelengths of the mth line of the Lyman series and the Balmer series of a hydrogen atom, we first need to understand some basic concepts.

1. Lyman series: The Lyman series consists of spectral lines of hydrogen that are obtained when an electron transitions from higher energy levels to the lowest energy level (n=1). The formula to calculate the wavelength of the mth line in the Lyman series is given by:

λ = R(1/1^2 - 1/m^2), where R is the Rydberg constant.

2. Balmer series: The Balmer series consists of spectral lines of hydrogen that are obtained when an electron transitions from higher energy levels to the second energy level (n=2). The formula to calculate the wavelength of the mth line in the Balmer series is given by:

λ = R(1/2^2 - 1/m^2), where R is the Rydberg constant.

Now, let's calculate the ratio:

Ratio = (wavelength of mth line in Lyman series) / (wavelength of mth line in Balmer series)

Substituting the formulas for the wavelengths:

Ratio = [R(1/1^2 - 1/m^2)] / [R(1/2^2 - 1/m^2)]

Simplifying the equation:

Ratio = [(1/1^2 - 1/m^2)] / [(1/2^2 - 1/m^2)]

Ratio = [(m^2 - 1) / m^2 ] / [(4 - 1) / 4]

Ratio = [4(m^2 - 1) / m^2] / 3

Therefore, the ratio of the wavelengths of the mth line of the Lyman series to the Balmer series is [4(m^2 - 1) / 3m^2].